Subgroups — #36368572

Order 22144.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26012414	C₂	1	2	yes	maximal
2	#26011870	C₂	1	1	yes	maximal, characteristic
3	#19149753	—	2	2	no	—
4	C₂×#14583816	—	2	2	no	—
5	#19149679	C₄	1	1	yes	characteristic
6	C₄×#13063436	C₂²	1	1	yes	characteristic
7	C₄⋊Q₈×C₁₇₃	C₄	1	1	yes	characteristic
8	#14583829	—	4	2	no	—
9	C₂×Dic₃₄₆	—	2	1	no	—
10	#14583816	—	4	2	no	—
11	#14583816	—	4	2	no	—
12	C₂×#13063436	D₄	1	2	yes	—
13	C₄⋊Dic₁₇₃	—	4	1	no	—
14	C₂×Q₈×C₁₇₃	—	2	1	no	—
15	C₄⋊C₄×C₁₇₃	—	4	1	no	—
16	C₄²×C₁₇₃	C₂×C₄	1	1	yes	characteristic
17	Dic₃₄₆	—	4	2	no	—
18	C₂×Dic₁₇₃	—	4	1	no	—
19	#13063436	D₈	1	2	yes	—
20	#13063436	SD₁₆	1	2	yes	—
21	Q₈×C₁₇₃	—	4	2	no	—
22	C₂×C₄×C₁₇₃	—	2	1	no	—
23	C₂×C₄×C₁₇₃	—	4	1	no	—
24	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₂²×C₁₇₃	#100	1	1	yes	characteristic
26	C₄×C₁₇₃	—	2	2	no	—
27	C₄×C₁₇₃	—	8	1	no	—
28	C₄×C₁₇₃	#102	1	2	yes	—
29	Dic₁₇₃	—	8	1	no	—
30	C₂×C₁₇₃	—	2	1	no	—
31	C₂×C₁₇₃	#331	1	1	yes	characteristic
32	C₁₇₃	#1358	1	1	yes	characteristic, complemented, Sylow
33	#1358	—	173	1	no	maximal, Sylow
34	#494	—	173	1	no	—
35	#364	—	173	2	no	—
36	C₂×Q₁₆	—	346	2	no	—
37	C₄⋊Q₈	—	173	1	no	—
38	C₄⋊Q₈	#10621924	1	1	yes	characteristic
39	M₃₂	—	346	2	no	—
40	C₄×C₈	—	173	1	no	—
41	C₂×Q₈	—	2	1	no	—
42	C₂×Q₈	—	346	1	no	—
43	Q₁₆	—	692	2	no	—
44	Q₁₆	—	692	2	no	—
45	C₂×C₈	—	173	2	no	—
46	C₄⋊C₄	—	4	1	no	—
47	C₄⋊C₄	—	692	1	no	—
48	C₄²	C₂×#10621924	1	1	yes	characteristic
49	C₁₆	—	692	2	no	—
50	Q₈	—	4	2	no	—
51	Q₈	—	692	2	no	—
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	4	1	no	—
54	C₂×C₄	—	692	1	no	—
55	C₂×C₄	C₂²⋊#10621924	1	1	yes	characteristic
56	C₈	—	173	2	no	—
57	C₈	—	173	2	no	—
58	C₂²	#19149739	1	1	yes	characteristic
59	C₄	—	2	2	no	—
60	C₄	—	8	1	no	—
61	C₄	—	1384	1	no	—
62	C₄	#19149741	1	2	yes	—
63	C₂	—	2	1	no	—
64	C₂	#26012349	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26012349
commutator	C₄²×C₁₇₃	C₂×C₄
Frattini	C₄²	C₂×#10621924
Fitting	C₄⋊Q₈×C₁₇₃	C₄
socle	C₂×C₁₇₃	#331
prime radical	C₂²×C₁₇₃	#100
cosocle	C₄×#13063436	C₂²
hypercenter	C₄⋊Q₈	#10621924
hypocenter	C₁₇₃	#1358
LCS	G ▷ C₄²×C₁₇₃ ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊Q₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
32. C₁₇₃	#1358	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#26012414	2
#26011870	2
#1358	173